Understand Bourgain-Chang's proof of bounds on exponential sums over small multiplicative subgroups.
Bourgain, J.(1-IASP); Chang, M.-C.(1-CAR) Exponential sum estimates over subgroups and almost subgroups of $\Bbb Z\sb Q\sp *$, where $Q$ is composite with few prime factors. Geom. Funct. Anal. 16 (2006), no. 2, 327--366.
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Speaker: Pär Kurlberg
Time/place: friday 070223, 14.15-15.15, room 3733.
Topics: Some classical bounds on complete exponential sums, such as Gauss and Kloosterman sums, the Weil bounds (aka the Riemann hypothesis for curves). Bounds on incomplete sums (Polya-Vinogradov and Burgess), and some applications (e.g. the size of the smallest quadratic non-residue.)
Speaker: Pär Kurlberg
Time/place: tuesday 070227, 10.15-12, room 3721.
Topic: We will start with an overview of the proof.
Speaker: Pär Kurlberg
Time/place: tuesday 070306, 10.15-12, room 3721.
Topic: We will wrap up the sketch of the proof by finishing the "statistical" multiplicative invariance. We will then go onto the proof of proposition 2.1.
Time/place: tuesday 070313, 10.15-12, room 3721.
Speaker: Pär Kurlberg
Topic: We will finish the proof of proposition 2.1 - how to use BGS' (Balog-Gowers-Szemeredi) to go from "statistical" multiplicative invariance to true multiplicative invariance.
Time/place: wednesday 070314, 13.00-15.00, room 3733.
Speaker: Alexander Engström
Topic: On addititive combinatorics and the proof of Balog-Gowers-Szemeredi.
Time/place: 10:15 - 12:00, Tuesday 20 March 2007, room 3721.
Speaker: Liangyi Zhao
Topic: I'll talk about bounds for character sums due to D. A. Burgess. In the interest of time, I will not go through all the details of the proof but present only the key points. Also, I will simply quote some results of H. Hasse and A. Weil which are needed in the proof.
Reference:
D. A. Burgess, On Characters Sums and Primitive Roots, Proc. London Math. Soc., No. 3, Vol. 12, 1962, pp. 179 - 192